In-situ stress fields and pore pressures are crucial for analyzing and predicting geomechanical issues encountered in the oil and gas industry. Drilling, completion, wellbore stability, fracturing the formation, etc. involve significant financial investment. Reservoir stress changes occurring during production, such as reservoir compaction, surface subsidence, formation fracturing, casing deformation and failure, sanding, or reactivation of faults may cause great loss. Therefore, better knowledge of the in-situ stress fields helps to reduce the losses and also contributes to better prediction and planning of the drilling and completion.
In general, the in-situ stress fields may be represented as a second-rank tensor with three principal stresses, namely the vertical stress (Sv), the minimum horizontal stress (Sh) and the maximum horizontal stress (SH). The vertical stress may be estimated from an integral of the density log, while the minimum horizontal stress may be estimated using a poroelastic equation or a frictional equilibrium equation.
Analytical and/or semi-analytical methods are used to characterize present day stress states in the sub-surface. These techniques are popular because they provide reasonable estimates of the stress distribution around and along the wellbore without building and solving a numerical grid, which saves a lot of time. Further, these techniques require only limited number of input parameters, which can be directly or indirectly observed by wireline tools or by specific tests done on core samples.
Although helpful, the assumptions and simplifications applied in these analytical solutions are not valid for all cases, and may lead to erroneous estimation of horizontal stresses. As an example, plain-strain solutions assume earth to be an elastic, homogenous and isotropic medium. Frictional equilibrium based calculations assume frictional strength of the faults as the limiting factors for the stresses, and allows stress estimations at limited number points with wellbore failures.
There is also the concern that in unconventional reservoirs, where the rock properties are not in conformation with already established models, reliable estimation of horizontal stresses for non-elastic rocks may be difficult to obtain.
For example, currently available analytical techniques to estimate horizontal stresses in the earth's crust use unrealistic assumptions and material models. Most of the analytical solutions in the industry assume a uniaxial, elastic, homogeneous and isotropic earth medium, which is not valid in the presence of structures such as faults, folds and also in the presence of plastic rocks such as ductile shale, etc.
Another approach uses frictional strength of the faults as the limiting case for stress estimation. Assumptions associated with this technique are more realistic than solutions with elasticity. However, the stress estimation based on this technique requires more input parameters. Stress calculations can be done at specific points along the wellbore where wellbore failures, such as breakouts and drilling-induced tensile fracture, are observed. This technique fails to provide stress estimation in the absence of wellbore failures. Also, this approach uses manual point based calculations that allow stress estimation only at a limited number of points and fails to produce a continuous estimation of stress along the borehole.
Analytical solutions for stress estimation for non-elastic medium are not developed because of the complexity and multi-dimensional nature of the problem. In fact, any non-elastic solution will need various assumptions. Also, this type of solution is only possible for simplified non-elastic materials.
As an example, most of the oil industry uses a plain-strain model to define a stress state, as illustrated below in Equation (1). The plain-strain approach assumes an elastic, homogenous and isotropic earth. It also assumes that the vertical stress (Sv) is applied instantaneously and that no other source of stress exists.
                                          S            Hmax                    -                      α            ⁢                                                  ⁢                          P              p                                      =                                            S              hmin                        -                          α              ⁢                                                          ⁢                              P                p                                              =                                    (                                                S                  v                                -                                  α                  ⁢                                                                          ⁢                                      P                    p                                                              )                        ⁢                          (                              v                                  1                  -                  v                                            )                                                          (        1        )            
where Pp is the pore pressure, α is Biot's coefficient, SH max and Sh min are horizontal stresses, and v is Poisson's ratio.
To account for existing tectonic stresses on the earth, Equation (1) is modified with stress and strain offset in the direction of tectonic forces. Equations (2) and (3) below represent the plain-strain models with stress and strain offsets respectively.
                                                        S              Hmax                        -                          α              ⁢                                                          ⁢                              P                p                                              =                                                    (                                  v                                      1                    -                    v                                                  )                            ⁢                              (                                                      S                    v                                    -                                      α                    ⁢                                                                                  ⁢                                          P                      p                                                                      )                                      +                          (                                                S                  y                                -                                  α                  ⁢                                                                          ⁢                                      P                    p                                                              )                                      ⁢                                  ⁢                                            S              hmin                        -                          α              ⁢                                                          ⁢                              P                p                                              =                                                    (                                  v                                      1                    -                    v                                                  )                            ⁢                              (                                                      S                    v                                    -                                      α                    ⁢                                                                                  ⁢                                          P                      p                                                                      )                                      +                          (                                                S                  x                                -                                  α                  ⁢                                                                          ⁢                                      P                    p                                                              )                                                          (        2        )            
where Sy and Sx are stress offsets due to tectonic movements in maximum and minimum horizontal stress directions respectively.
                                                        S              Hmax                        -                          α              ⁢                                                          ⁢                              P                p                                              =                                                    (                                  v                                      1                    -                    v                                                  )                            ⁢                              (                                                      S                    v                                    -                                      α                    ⁢                                                                                  ⁢                                          P                      p                                                                      )                                      +                                          E                                  (                                      1                    -                                          v                      2                                                        )                                            ⁢                              (                                                      ɛ                    H                                    +                                      v                    ⁢                                                                                  ⁢                                          ɛ                      h                                                                      )                                                    ⁢                                  ⁢                                            S              hmin                        -                          α              ⁢                                                          ⁢                              P                p                                              =                                                    (                                  v                                      1                    -                    v                                                  )                            ⁢                              (                                                      S                    v                                    -                                      α                    ⁢                                                                                  ⁢                                          P                      p                                                                      )                                      +                                          E                                  (                                      1                    -                                          v                      2                                                        )                                            ⁢                              (                                                      ɛ                    h                                    +                                      v                    ⁢                                                                                  ⁢                                          ɛ                      H                                                                      )                                                                        (        3        )            
where E is static Young's modulus, and εH and εh are tectonic strains in maximum and minimum horizontal stress directions respectively.
Recently, Equation (3) was modified to consider transverse anisotropy in a shaly medium, which constitutes most of the non-conventional reservoirs. Equation (4) shows a plain-strain model for a transversely anisotropic medium.
                                                        S                              H                ⁢                                                                  ⁢                max                                      -                                          α                h                            ⁢                              P                p                                              =                                                    (                                                      E                    h                                                        E                    v                                                  )                            ⁢                              (                                                      v                    v                                                        1                    -                                          v                      h                                                                      )                            ⁢                              (                                                      S                    v                                    -                                                            α                      v                                        ⁢                                          P                      p                                                                      )                                      +                                                            E                  h                                                  (                                      1                    -                                          v                      h                      2                                                        )                                            ⁢                              (                                                      ɛ                    H                                    +                                                            v                      h                                        ⁢                                          ɛ                      h                                                                      )                                                    ⁢                                  ⁢                                            S                              h                ⁢                                                                  ⁢                min                                      -                                          α                h                            ⁢                              P                p                                              =                                                    (                                                      E                    h                                                        E                    v                                                  )                            ⁢                              (                                                      v                    v                                                        1                    -                                          v                      h                                                                      )                            ⁢                              (                                                      S                    v                                    -                                                            α                      v                                        ⁢                                          P                      p                                                                      )                                      +                                                            E                  h                                                  (                                      1                    -                                          v                      h                      2                                                        )                                            ⁢                              (                                                      ɛ                    h                                    +                                                            v                      h                                        ⁢                                          ɛ                      H                                                                      )                                                                        (        4        )            
where subscripts h and v represent the values in vertical and horizontal directions respectively.
Another approach to define stress states in the earth is the frictional equilibrium approach used by GMI in the SFIB tool kit (geomi.com/software/SFIB.php). This approach assumes that the earth is full of discontinuities (faults and fractures) and these discontinuities control the maximum value of stress a block of earth can hold. It uses borehole failures such as breakouts and tensile fractures to define the stress state. This approach is the other end of the spectrum than a plain-strain model. The equation of frictional equilibrium state is shown in Equation (5).
                                          σ            1                                σ            3                          =                                                            S                1                            -                              α                ⁢                                                                  ⁢                                  P                  p                                                                                    S                3                            -                              α                ⁢                                                                  ⁢                                  P                  p                                                              ≤                                    [                                                                    (                                                                  μ                        2                                            +                      1                                        )                                                        1                    /                    2                                                  +                μ                            ]                        2                                              (        5        )            
where S1 and S3 are the maximum and minimum principal stresses, and μ is the coefficient of frictional strength of faults and fractures in the medium.
Plain-strain model in the above forms (Equations 1 to 4) are used extensively in the oil industry, but fail to account for the fundamental reality that the earth is not elastic and homogenous. The frictional equilibrium approach (Equation 5) is a better approach to get the stress magnitudes in the presence of borehole failures and to get the maximum threshold of stresses in the earth. However, it doesn't explain the stress state before the borehole failures, or how stresses are affected by the non-elastic nature of the rock.
Therefore, there is the need for a better method of estimating horizontal stress that takes both the frictional strength and realistic elasticity into consideration.